Journal of Jilin University(Engineering and Technology Edition) ›› 2026, Vol. 56 ›› Issue (3): 758-771.doi: 10.13229/j.cnki.jdxbgxb.20240914

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Comparative of analytical solutions for axial symmetric dynamic response of asphalt pavements

Hua-yu SHANG1,2(),Hai-shan FAN1,2,Jun-hui ZHANG1,2()   

  1. 1.Key Laboratory for Highway Engineering of Ministry of Education，Changsha University of Science and Technology，Changsha 410114，China
    2.School of Transportation，Changsha University of Science& Technology，Changsha 410114，China
  • Received:2024-08-19 Online:2026-03-01 Published:2026-03-31
  • Contact: Jun-hui ZHANG E-mail:shy@stu.csust.edu.cn;zjhseu@csust.edu.cn

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Abstract:

The current researches on pavement mechanics usually focus on the derivation of analytical solutions and the exploration of the factors affecting them， there are few studies on the difference of analytical solutions. In order to solve this problem， transfer matrix method， stiffness matrix method， wave transfer method and reflection and transmission matrix method were applied in the deduction process of the analytical solutions considering the transverse isotropy of materials and the contact state between layers. On this basis， the calculation efficiency and the range of adaptability of the above four analytical solutions are studied by combining four kinds of pavement structures. The results show that： ①for the matrix scale， reflection and transmission matrix method， transfer matrix method， stiffness matrix method and wave transfer method increase in turn； ② for the solution speed， the transfer matrix method takes much less time than the other three methods， while the reflection and transmission matrix method consumes the most time； ③in the numerical stability， the transfer matrix method is very easy to overflow data when calculating the dynamic response of finite thickness， while the other analytical solution methods have good numerical stability； ④in practical application， the transfer matrix method is particularly suitable for the surface of semi space problems， and the stiffness matrix method can only calculate the dynamic response of the surface of each structural layer， however， the wave transfer method and the reflection and transmission matrix method can solve the dynamic response of any position in the pavement.

Key words: mechanics of the layered system, transfer matrix method, stiffness matrix method, wave transfer method, reflection and transmission matrix method

CLC Number: 

  • U416.2

Fig.1

Schematic diagram of n-layer layered system"

Table 1

Material Parameters of Pavement Layers"

结构层Ev/MPanμv（=μh）ρ/（kg?m-3）h/mαx
路面7 0000.30.252 4000.180.8
基层10 0000.50.302 3000.350.0
底基层4001.00.352 2000.200.0
土基1001.00.351 6003.00-

Fig.2

Comparison diagrams of calculation results of different analytical solutions （χ=300， ΔL=5）"

Fig.3

Research on deflection curve convergence （stiffness matrix method）"

Fig.4

Research on horizontal strain curve convergence （stiffness matrix nethod）"

Fig.5

Research on deflection peak convergence （stiffness matrix method）"

Fig.6

Comparison of literatures"

Fig.7

Schematic diagram of pavement structure"

Table 2

Calculative time of different analytical solutions"

计算方法路面结构传递/反射透射矩阵/矩阵元素计算构建矩阵方程并求解数值逆变换总耗时

传递矩

阵法

A76.354.428.9160.9
B计算失败，计算结果为NAN
C80.254.728.9191.1
D计算失败，计算结果为NAN

刚度矩

阵法

A423.4121.330.2590.7
B398.2116.827.9557.3
C554.4141.430.1741.5
D531.5138.627.8712.8

波传

递法

A123.2396.631.0643.1
B113.5401.828.2629.1
C158.4496.429.7821.7
D147.7499.728.0803.9

反射、

透射

矩阵法

A76.8555.130.5677.8
B70.4559.229.1674.3
C101.3745.431.6894.1
D95.5756.129.1895.8

Fig.8

Deflection of different pavement structures"

Table 3

Characteristics and application scope of different analytical solutions"

计算方法

计算

速度

数值

稳定性

求解位置适应范围
传递矩阵法快较稳定路面表面半空间
刚度矩阵法一般稳定各结构层表面

半空间

有限厚度

波传递法较慢稳定任意位置

半空间

有限厚度

反射、透射矩阵法较慢稳定任意位置

半空间

有限厚度

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